Input-output formalism

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Table of contents

Introduction
1 Introduction to opto- and electromechanics 
1.1 Thermal motion of a harmonic oscillator
1.1.1 Equation of motion of a mode
1.1.2 The autocorrelation function
1.1.3 Noise spectral density
1.1.4 The uctuation-dissipation theorem
1.1.5 Noise spectrum of the harmonic oscillator
1.2 Optical measurement of motion
1.2.1 Position-dependent phase shift
1.2.2 The Mach-Zehnder interferometer
1.3 Cavity electromechanics
1.3.1 The intracavity light eld of a bare circuit
1.3.2 Input-output formalism
1.3.3 Dispersive coupling
1.3.4 Dynamical backaction
1.3.5 Electromechanical cooling
1.4 Summary
2 First electromechanical experiments 
2.1 Silicon-nitride membrane electromechanics
2.1.1 Choosing the mechanical resonator
2.1.2 Electromechanical devices with SiN membranes
2.1.3 Fabricating the electromechanical device
2.2 Characterization experiments
2.2.1 Characterizing the cavity resonance
2.2.2 Optomechanically Induced Transparency (determining m)
2.2.3 Measuring g0
2.2.4 Ringdown measurement of 􀀀m
2.2.5 Summary
2.3 Cooling experiments
2.4 Discussion
2.4.1 The microwave cavity
2.4.2 The mechanical resonator
2.5 Summary
3 Membrane design and simulations 
3.1 Key parameters of SiN nanomembranes
3.1.1 Quality factor of a harmonic oscillator
3.1.2 Mode prole of a vibrating plate
3.1.3 Bending losses
3.1.4 Radiation losses
3.1.5 Scaling of the parameters with the membrane geometry
3.2 Phononic crystal membrane design
3.2.1 Engineering bandgaps in periodic structures
3.2.2 Localized defect states in PnC membranes
3.2.3 The curvature prole of defect and edge modes
3.2.4 Computing Qb of D1 and VE1
3.2.5 A \nal » consistency check
3.3 Mode coupling in PnC membranes
3.3.1 Coupling of lossy harmonic oscillators
3.3.2 Numerical analysis of edge mode coupling
3.3.3 Edge mode engineering
3.3.4 Double-defect membranes
3.4 Concluding remarks
4 Membrane fabrication and characterization 
4.1 Fabrication procedure
4.1.1 Wafer details
4.1.2 Releasing free-standing PnC membranes
4.1.3 Additional fabrication details
4.2 Experimental Setup
4.2.1 The vacuum chamber
4.2.2 A shot-noise limited optical interferometer
4.2.3 Driving the mechanical motion
4.3 Experimental methods
4.3.1 Measuring the spatially-dependent thermal spectrum
4.3.2 Measuring the mode prole
4.3.3 Ringdown measurement of the quality factor
4.4 Results for defect and edge modes
4.4.1 Measured thermal spectra
4.4.2 Measured mode proles
4.4.3 Measured Qb
4.5 Results for dimer membranes
4.5.1 Measured thermal spectra and mode proles
4.5.2 Determining the dimer coupling rate
4.6 Concluding remarks
5 Conclusion and outlook 
5.1 Single-defect PnC membranes
5.2 Preparing non-Gaussian states of motion
5.2.1 Direct coupling
5.2.2 Extrinsic nonlinearity
5.2.3 Counting single microwave photons
5.3 Concluding remarks
Appendix A Calibrating the resonator population 
Appendix B Scaling of the eective population 
Appendix C Coupled damped harmonic oscillators 
C.1 Eigenfrequencies
C.2 Eigenvectors
Appendix D Fabrication recipes 
D.1 Plain SiN membrane fabrication
D.2 Patterned SiN membrane fabrication
Appendix E Symbol list